Problem 18 - Algebra and Trigonometry

Question

\(3-24=A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular coordinate equation for the curve by eliminating the parameter. $$ x=\cos 2 t, \quad y=\sin 2 t $$

Step-by-Step Solution

Answer

(a) The curve is a unit circle. (b) Rectangular equation: \( x^2 + y^2 = 1 \).

Step 1: Understand the Parametric Equations

The given parametric equations are:\[ x = \cos(2t) \quad \text{and} \quad y = \sin(2t) \] These represent the x and y coordinates of points on the curve as functions of the parameter \( t \). Each point on the curve corresponds to a different value of \( t \).

Step 2: Sketch the Curve

Since the parametric equations use sine and cosine, they describe a circle or an ellipse in the xy-plane. Here, \( x^2 + y^2 = \cos^2(2t) + \sin^2(2t) = 1 \) holds for all \( t \). Thus, the parametric equations correspond to the unit circle\( \{(x, y): x^2 + y^2 = 1 \} \), but traced differently with respect to \( t \).

Step 3: Eliminate the Parameter

To convert to a rectangular coordinate form, use the Pythagorean identity:\[ \cos^2(2t) + \sin^2(2t) = 1 \] Equation rearrangement gives \( x^2 + y^2 = 1 \). Hence, the rectangular equation of the curve is:\[ x^2 + y^2 = 1 \]

Key Concepts

Rectangular Coordinate Equation

To understand what a rectangular coordinate equation is, let's explore its significance. A rectangular coordinate equation describes the relationship between the coordinates \( x \) and \( y \) on a two-dimensional plane, often known as the Cartesian plane. This form of equation eliminates any parameter involved in parametric equations to present a clear, direct relationship between \( x \) and \( y \).

In the context of the given exercise, we start from the parametric equations \( x = \cos(2t) \) and \( y = \sin(2t) \). These equations show how \( x \) and \( y \) relate to a parameter \( t \). To convert these to a rectangular form, we apply trigonometric identities that involve only \( x \) and \( y \), thereby removing the parameter \( t \).

The result, \( x^2 + y^2 = 1 \), is a simple and classical example of a rectangular equation, representing a circle on the xy-plane. This process is crucial in many fields of mathematics and engineering, where understanding the direct relations between coordinates is more practical.

Unit Circle

The term "unit circle" refers to a circle with a radius of one, centered at the origin in the coordinate plane. This is an essential concept in trigonometry and helps us understand the correlation between angles and lengths on a circle. The equations \( x = \cos(\theta) \) and \( y = \sin(\theta) \) form the unit circle, where \( \theta \) is the angle in radians.

In the parametric equations \( x = \cos(2t) \) and \( y = \sin(2t) \), the values of \( x \) and \( y \) describe points on the unit circle as \( t \) varies. Usually, \( x^2 + y^2 = 1 \) confirms the curve is a unit circle. This is due to the Pythagorean identity \( \cos^2\theta + \sin^2\theta = 1 \), which holds true for any angle \( \theta \). By sketching, you see how these trigonometric functions traverse the circle.

The unit circle is fundamental for visualizing trigonometric functions, proving identities, and solving equations involving angles and lengths.

Pythagorean Identity

The Pythagorean identity is a crucial principle in trigonometry, named after the Greek mathematician Pythagoras. It expresses a fundamental relationship between the sine and cosine functions for a given angle: \( \cos^2\theta + \sin^2\theta = 1 \). This identity is essential for converting parametric equations into rectangular form.

In our example, the parametric equations \( x = \cos(2t) \) and \( y = \sin(2t) \) satisfy the Pythagorean identity by forming \( x^2 + y^2 = 1 \). This expression confirms that the points lie on the unit circle. By applying this identity, we can eliminate the parameter \( t \) and understand the inherent geometric shape, which is important for both theoretical mathematics and practical problem-solving.

This identity is not only fundamental for proving various trigonometric properties but also for simplifying expressions and solving complex trigonometric equations. Understanding and applying the Pythagorean identity is a stepping stone to mastering trigonometry.